Volume Stability for Hyperbolic Manifolds and Applications to General Relativity
arXiv:2607.27666
The paper proves that for closed hyperbolic 3‑manifolds, any sequence of metrics with scalar curvature ≥ −6 whose volumes approach the hyperbolic volume must converge in the C⁰ sense to the hyperbolic metric away from regions of vanishing volume, and applies this result to show stability of the Fischer–Moncrief reduced Hamiltonian in vacuum general relativity.
Abstract
We prove a sharp volume-stability theorem for closed hyperbolic three-manifolds. Let \((M,h)\) be closed hyperbolic with \(\operatorname{Ric}_h=-2h\), and let \(g_i\) be smooth metrics on \(M\) satisfying . After passing to a subsequence, there exist \(Z_i\subset M\), smooth domains \(K_i\subset M\), and diffeomorphisms such that and Thus near-equality in the sharp hyperbolic volume bound forces tensorial \(C^0\)-convergence to the hyperbolic metric outside regions of vanishing volume. As an application, we establish stability of the Fischer--Moncrief reduced Hamiltonian at the Lorentz-cone ground state: after CMC normalization, near-minimizing compact vacuum data in the hyperbolic topological class converge, modulo sets of vanishing volume, to the hyperbolic Lorentz-cone geometry in tensorial \(C^0\). This provides a rigorous volume-dominance formulation of the Fischer--Moncrief asymptotic picture.
36 pages, comments welcome